Caustic free completion of pressureless perfect fluid and k-essence
CCaustic free completion of pressureless perfect fluid and k-essence
Eugeny Babichev a,b ∗ and Sabir Ramazanov c a Laboratoire de Physique Th´eorique, CNRS,Univ. Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France b UPMC-CNRS, UMR7095, Institut d’Astrophysique de Paris, G R ε C O , 98bis boulevard Arago, F-75014 Paris, France c Gran Sasso Science Institute (INFN), Viale Francesco Crispi 7, I-67100 L’Aquila, Italy
September 20, 2018
Abstract
Both k-essence and the pressureless perfect fluid develop caustic singularities atfinite time. We further explore the connection between the two and show that they be-long to the same class of models, which admits the caustic free completion by means ofthe canonical complex scalar field. Specifically, the free massive/self-interacting com-plex scalar reproduces dynamics of pressureless perfect fluid/shift-symmetric k-essenceunder certain initial conditions in the limit of large mass/sharp self-interacting po-tential. We elucidate a mechanism of resolving caustic singularities in the completepicture. The collapse time is promoted to complex number. Hence, the singularityis not developed in real time. The same conclusion holds for a collection of collision-less particles modelled by means of the Schroedinger equation, or ultra-light axions(generically, coherent oscillations of bosons in the Bose–Einstein condensate state).
Theories with non-canonical kinetic terms are of particular interest in cosmology and inthe field of modified gravities. These include k-essence [1, 2, 3, 4], ghost condensate [5]and Generalized Galileons, or Horndeski, models [6, 7, 8, 9, 10, 11]. See the review [12] ∗ e-mails : [email protected], [email protected] a r X i v : . [ h e p - t h ] S e p or a complete list. Admitting such non-trivial field theories opens up new prospectives foraddressing standard cosmological problems, e.g., the smallness of Λ-constant and the initialsingularity problem.At the same time, theories of that kind, even free of any obvious pathologies, may secretlypossess some unappealing properties. First, they typically exhibit the sub-/superluminalityand, therefore, look quite uncommon for a particle physicist. This is, e.g., the case of k-essence/Generalized Galileons. While superluminality does not immediately entail causalparadoxes [13], it may nevertheless obstruct the UV-completion by means of a local Lorentz-invariant quantum field theory [14].Another shortcoming of k-essence has been revealed recently [20]: it leads to the appear-ance of caustic singularities (see also Refs. [21, 22, 23]). Namely, characteristics of equationsof motion cross at some finite time, what results into multivalued derivatives of the k-essencefield. This has been proved in Ref. [20] for the case of the generic simple wave. The sameconclusion holds in the class of Generalized Galileon models involving k-essence.Appearance of caustics brings together k-essence and the pressureless perfect fluid (PPF).Typically employed as an approximate description for the collection of collisionless particles,PPF also develops singularities. There is no actual problem from the particle physics pointof view: shell-crossing merely signals the breakdown of the fluid description. Namely, asingularity is avoided by allowing the multi-stream regime. However, PPF is more genericand may arise as the field-theoretical construction in some modifications of gravity, e.g.,scalar Born–Infeld theories [15], mimetic matter scenario [16, 17] and the projectable versionof the Horava–Lifshitz gravity [18, 19]. In that case, one should design another way of curingsingularities.In the present paper, we point out an even deeper connection between the shift-symmetrick-essence and PPF. We show that they belong to the same class of models, which admitsthe caustic free completion by means of the canonical complex scalar field, see Section 2.In particular, PPF corresponds to the free massive field. On the other hand, the genericpotential with self-interactions leads to k-essence.In what follows, we restrict to the classical level analysis. Therefore the caustic freecompletion discussed here should not be confused with the quantum field theoretical UV-completion. Even within a purely classical approach, the correspondence between the k-essence/PPF and the complex scalar is not straightforward. This clearly follows from thedegree of freedom (DOF) counting: only one DOF is enough to describe dynamics of k-essence/PPF, and two DOFs are required in the case of the complex scalar. The problemcan be addressed by a proper choice of the initial configuration for the complex field, as isshown in Fig. 1. Modulo the cosmological drag, we set its amplitude to the constant valuein the early Universe. This requirement fixes the frequency dependence of the field andeffectively eliminates one extra degree of freedom.2sing a particular example of PPF, we show how the free massive complex field re-produces its dynamics. Both follow the same evolution down to the times, when causticsingularities start to be formed. Since this point on, the discrepancy between two scenariosbecomes unavoidable: while the description in terms of PPF breaks down, the actual sin-gularity does not emerge in the complete picture. In the latter case, the collapse time ispromoted to the complex quantity. Hence, the real time evolution always remains smooth.While we deduce this conclusion from the study of PPF evolution, we conjecture that thesame mechanism is generic and also works for k-essence.Note that the similarity between the complex scalar dynamics and superfluids is well-known [24, 25]. In the context of k-essence models, the correspondence was pointed out inRefs. [26, 27]. However, the idea has been barely used in the field of modified gravity toaddress shortcomings of k-essence. We fill in this gap in the present work.Furthermore, PPF can be modeled by means of the complex field—quantum mechan-ical wave function obeying Schroedinger equation [28, 29, 30, 31, 32, 33, 34]. This obser-vation is often used as an efficient tool to study the dynamics of collisionless dust parti-cles without resorting to cumbersome N-body simulations. Notably, in some situations ofinterest in cosmology, the Schroedinger equation provides the genuine description of thephysical system. This is, e.g., the case of ultra-light axions at distances smaller thantheir de Broglie wavelength, or, more generally, bosons in the Bose-Einstein condensatestate [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49] . Results of the present workcan be readily applied to all those cases.The outline of the paper is as follows. In Section 2, building on the proposal of Ref. [20],we consider the class of models, which comprises k-essence and PPF. In Section 3, we elab-orate the conditions, under which the free massive scalar complex field reproduces the dy-namics of PPF. There we also discuss the mechanism of the caustic avoidance. In Section 4,we generalize the conclusions made in the context of PPF to k-essence. We finalize withsome discussions in Section 5. We start with the action given by [20], S = (cid:90) d x √− g (cid:20) (cid:15) ∂ µ λ ) + λ ∂ µ ϕ ) − V ( λ ) (cid:21) . (1)Here λ and ϕ are scalar fields, and (cid:15) is the dimensionful constant. Note that Ref. [20] dealswith the kinetic term ∼ λ ( ∂ µ ϕ ) for the field ϕ . With our choice (1), the kinetic term Strictly speaking, the Schroedinger equation is suitable for describing the non-interacting axion. Onceself-interactions are included into the analysis, it must be replaced by the Gross–Pitaevskii equation.
3s manifestly positively defined, and we avoid any possible issues with ghost instabilities.Equations of motion following from the action (1) are given by (cid:15) (cid:3) λ − λ ( ∂ µ ϕ ) + V (cid:48) ( λ ) = 0 . (2)and ∂ µ ( √− gλ ∂ µ ϕ ) = 0 . (3)Upon setting (cid:15) = 0, Eq. (2) reduces to an algebraic equation, which can be used to expressthe variable λ as the function of X ≡ g µν ∂ µ ϕ∂ ν ϕ , i.e., λ = F ( X ). Substituting this backinto the action, we reproduce the shift-symmetric k-essence action, S = (cid:90) d x √− g L ( X ) , where one should identify (cid:112) L (cid:48) ( X ) = F ( X ).Note that the model (1) contains not just the k-essence. Indeed, consider the quadraticpotential V ( λ ) = λ . In that case, the field λ plays the role of the Lagrange multiplierand, hence, cannot be expressed as the function of X . Therefore, this case does not matchany of k-essence scenarios. Still, it represents a physically relevant situation. Indeed, thestress-energy tensor for the choice V ( λ ) = λ is given by, T µν = λ ∂ µ ϕ∂ ν ϕ . We recognize the pressureless perfect fluid (PPF) described by the energy density λ and thevelocity potential ϕ [50]. PPF is perhaps the best known example of the system developingcaustic singularities. We conclude that the k-essence and PPF indeed represent particularexamples of one and the same model.Switching to the case of the non-zero parameter (cid:15) , i.e., (cid:15) (cid:54) = 0, promotes the field λ tothe dynamical degree of freedom. Let us elucidate the physical content of the model in thatcase. For this purpose, it is convenient to redefine the variables, √ (cid:15)λ = ˜ λ ˜ ϕ = ϕ √ (cid:15) , and to introduce the complex scalar fieldΨ ≡ Ψ + i Ψ = ˜ λ · e i ˜ ϕ . (4)It is easy to see that the action (1) can be recast in the following simple form, S = (cid:90) d x √− g (cid:20) | ∂ µ Ψ | − V (Ψ) (cid:21) . (5)Remarkably, we arrive at the action of the canonical complex scalar field.4onsider the renormalizable potential V of the form, V (Ψ) = α · M | Ψ | βM | Ψ | , (6)where we introduced the notation M ≡ (cid:15) ; α , β and Λ are some arbitrary constants. We see that the limit of infinitely small (cid:15) corre-sponds to infinitely heavy mass of the field Ψ and/or a very steep potential (for the fixedparameters α , β and Λ).Note in passing that the complex scalar with the potential (6) often arises in the contextof Bose-Einstein condensates. That is, for a sufficiently large de Broglie wavelength and/orat large densities, the classical approach breaks down. In that regime, excitations of bosonsare described by the complex field satisfying the Schroedinger equation (if bosons are non-interacting) or the Gross–Pitaevskii equation (provided that there is the self-interaction).The action (5) can be viewed as the relativistic completion of the Bose–Einstein condensate.In what follows, we choose to do not concretize the physical origin of the field Ψ.Let us list the models associated with different values of the parameters α and β . • Setting β = 0 we get a model of a free massive complex field. As it has been explainedpreviously, this case corresponds to PPF. • The spontaneous symmetry breaking potential, i.e., with α < β >
0, stands forthe simplest subluminal k-essence scenario ∼ X + X , [26, 27]. • The unbounded potential, i.e., α > β <
0, corresponds to the superluminalk-essence model ∼ X − X , [26, 27]. The instability can be avoided, if we add higherpowers of the field Ψ. • Finally, for α > β > ∼ ( X − [26, 27],—ghostcondensate. Not surprisingly, caustic singularities have been identified in this contextas well [51].Needless to say, the canonical complex scalar is free of caustic singularities. Nevertheless,it is unclear, how the dynamics of PPF or k-essence is reproduced in this context. Thequestion is non-trivial, since dynamics of the complex scalar is described by two degrees offreedom, while the evolution of k-essence and PPF is fixed by one. Hence, the correspondenceholds only for the specific initial configuration of the complex field.To paraphrase, the model of the complex scalar does not provide with the genuine com-pletion of k-essence/PPF, but rather approximates the latter under certain conditions. Theaccuracy of the approximation grows with the mass parameter M . So, in the limit M → ∞ In the bulk of the present paper, we will deal with PPF. The reason is that it correspondsto the tractable case of the free massive scalar. Still, the main statements formulated in thisSection appear to be generic and can be extrapolated to the situation with the self-interactingscalar and, hence, k-essence.We will focus on two limiting cases: i) the homogeneous evolution of the complex scalaris dominated by the cosmological expansion; ii) inhomogeneities of the field Ψ are large, sothat the cosmic drag can be neglected. To simplify our analysis, we will switch off metricperturbations. This is justified, since caustic singularities represent the intrinsic property ofPPF, i.e., they occur even in the absence of gravity. Other simplifications will be discussed,where relevant.
In the homogeneous case, the equation of motion for the free massive field Ψ (we set α = 1and β = 0 in the potential (6)) is given by,¨Ψ + 3 H ˙Ψ + M Ψ = 0 . (7)The solution to this equation readsΨ = Aa / e iMt + Ba / e − iMt ; (8)here A and B are some constant amplitudes. More precisely, the solution (8) satisfies Eq. (7)modulo terms suppressed by the ratio H M , where H is the Hubble rate. For the genericamplitudes A and B , the evolution of the field Ψ is represented by a peculiar curve in the This curve is an ellipse in the limit of the static Universe. V ( ) V ( ) Figure 1: Complex field Ψ potential (6) for the choice of parameters ( α > , β = 0) (freemassive case, left) and ( α < , β >
0) (potential with spontaneous symmetry breaking,right). The former corresponds to PPF, while the latter stands for the simplest model ofk-esseence with sub-luminality,
L ∝ X + X . Dynamics of PPF and k-essence is reproducedfor the particular configuration of the complex field depicted by the dashed line.configuration space. The curve reduces to the circle in the particular case, B = 0 . (9)(alternatively, one could set A = 0), see Fig. 1. The relevance of that condition is clear. Onceit is imposed, the amplitude of the field Ψ is a slowly varying function (constant modulocosmic drag). Hence, one can neglect the first term on the l.h.s. of Eq. (2), which reducesto the constraint equation ∂ µ ϕ∂ µ ϕ = 1. The latter describes the geodesics motion of dustparticles [50]. In other words, we deal with PPF. Alternatively, one can calculate the pressure P and show that it equals to zero in the limit H (cid:28) M . Consistently, the energy density ρ ( t ) = | ˙Ψ | M | Ψ | . redshifts with the scale factor as ρ ( t ) ∼ a in the same limit. In the opposite situation, whenthe Hubble rate is large compared to the mass M , the equation of state is that of the stiffmatter [48], i.e., ρ = P . We will not be interested in those early times, however.We conclude that PPF is indeed reproduced from the massive complex scalar field upontuning the initial conditions. 7 .2 Inhomogeneous evolution Let us now switch to the case of our primary interest—inhomogeneous evolution of PPFand the complex scalar. Our goal is to specify conditions, which bring together these twoseemingly different models. It is convenient to work with the complex field representation interms of the normalized amplitude ˜ λ and the phase ˜ ϕ defined by Eq. (4). The time derivativeof the field Ψ is then given by, ˙Ψ = ∂ ln ˜ λ∂t Ψ + i · ∂ ˜ ϕ∂t Ψ . We can express the time derivative of the phase ˜ ϕ as follows, ∂ ˜ ϕ∂t = Im ˙ΨΨ . The quantity ∂ ˜ ϕ∂t including its inhomogeneous perturbation is not arbitrary in the case ofPPF but is defined by the constraint equation, i.e., ∂ ˜ ϕ∂t = ± (cid:112) M + ( ∂ i ˜ ϕ ) . (10)In what follows we stick to the plus sign on the r.h.s. This corresponds to the PPF velocitydefined as v i = − ∂ i ϕ . Therefore, the initial condition for the time derivative of the complexscalar cannot be arbitrary (if we are willing to reproduce PPF) but is fixed to be,Im ˙ΨΨ = (cid:112) M + ( ∂ i δ ˜ ϕ ) . (11)Let us argue that this condition is automatically satisfied in the limit of large M , i.e., M → ∞ .Recall that the condition (9) should be obeyed in the homogeneous case. This fixes thegeneric solution for the complex scalar to be of the form,Ψ = (cid:90) dkα ( k ) e ikx + i √ k + M t . (12)Using the latter, one writes for the time derivative of the field Ψ,˙Ψ = i (cid:113) − ∂ i + M Ψ , (13)The operator (cid:112) − ∂ i + M is defined by its Taylor expansion, (cid:113) − ∂ i + M = M (cid:32) (cid:88) n ( − n α n ∂ ni M n (cid:33) , α n are the coefficients of the expansion. Their precise values will not be relevant forus. For simplicity, consider the case n = 1. We have, ∂ i Ψ = (cid:32) ∂ i ˜ λ ˜ λ + 2 i∂ i ˜ λ ˜ λ ∂ i ˜ ϕ + i∂ i ˜ ϕ − ( ∂ i ˜ ϕ ) (cid:33) Ψ . The last term in brackets on the r.h.s. is the most relevant one, as it involves the secondpower of the phase ˜ ϕ . The latter is a large quantity. This follows from the chain of equalities ∂ i ˜ ϕ = M ∂ i ϕ = − M v . Hence, for the fixed velocity v , the quantity ˜ ϕ grows as M , andthe term ( ∂ i ˜ ϕ ) indeed dominates in the large M limit. At least, this is true, whenever thespatial distribution of the amplitude ˜ λ and the phase ˜ ϕ is sufficiently smooth, i.e., (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ i ˜ λ ˜ λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ L − (cid:28) M v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ i ˜ λ ˜ λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ L − (cid:28) M v | ∂ i ˜ ϕ | ∼ M vL (cid:28) M v . (14)Here L is the characteristic scale of inhomogeneities in the amplitude ˜ λ and the phase ˜ ϕ . If M − is some microscopic scale, that condition is satisfied with an excess in most situationsof interest in cosmology and astrophysics. Consequently, we get ∂ i Ψ = − ( ∂ i ˜ ϕ ) Ψ ( M → ∞ ) . The generalization to the case of arbitrary n is straightforward,( − n ∂ ni Ψ = ( ∂ i ˜ ϕ ) n Ψ ( M → ∞ ) . (15)We conclude that (cid:113) − ∂ i + M Ψ = (cid:112) ( ∂ i ˜ ϕ ) + M Ψ ( M → ∞ ) . Substituting this into Eq. (13), we see that Eq. (11) is indeed satisfied in the large M limit.To summarize: The negative-frequency branch of the generic solution of the free complexscalar field reproduces PPF in the limit of large M , provided that the distribution of thefields ˜ λ and ˜ ϕ is sufficiently smooth in space, i.e., the inequalities (14) are obeyed. Thatconclusion could be anticipated from the simpler considerations of the degree of freedomcounting. Indeed, PPF is described by one degree of freedom and, hence, is solved by twoinitial conditions. Consistently, the negative-frequency branch of the field Ψ is defined byone complex amplitude α ( k ), which is once again fixed by two real functions on the initialCauchy surface.As the inhomogeneities in the field ˜ λ grow, the inequalities (14) become progressively lessaccurate and so the correspondence between the complex scalar and PPF. The discrepancygets particularly strong close to the times, when the caustic singularity is supposed to beformed. This is basically the mechanism of completing PPF by means of the complex scalar.Soon, we will give a support in favor of this picture. Before that, let us establish theconnection with another closely related way of completing PPF.9 .3 Non-relativistic limit: connection to Schroedinger equation Since this point on and until the end of the Section, we switch to the non-relativistic limit.The solution for the complex scalar field then takes the form,Ψ = e iMt ˜Ψ ˜Ψ = (cid:90) dkα ( k ) e ikx + i k M t , where we readily dropped the positive-frequency part of the solution. It is straightforwardto see that the function ˜Ψ satisfies the Schroedinger equation, i ∂ ˜Ψ ∂t = ∂ M ∂x ˜Ψ . (16)Of course, its appearance is not a surprise, as the Klein-Gordon equation has been originallydesigned as the relativistic completion of the Schroedinger equation.The possibility to complete PPF by means of the Schroedinger equation is quite well-known in the literature [28, 29, 30, 31, 32, 33, 34]. So, it is often used to model a collectionof collisionless particles. This may be relevant for the study of the gravitational clustering,—a complicated process, which requires running cumbersome N-body simulations. There isalso one realistic situation, when the Schroedinger equation arises as the genuine completionof PPF. This is the case of ultra-light axions [39, 45, 49]. Namely, when the de Brogliewavelength of the axion is larger than the distance between the particles, the description interms of the wave function becomes more adequate.To set a connection between PPF and the quantum mechanical wave function, one per-forms the so called Madelung transformation,˜Ψ = λM e iMδϕ , where we made use of the ’non-canonical’ amplitude λ and the phase ϕ —most relevant forthe case of PPF. In terms of λ and ϕ , the Schroedinger equation can be equivalently writtenas the system of coupled equations, ∂ v ∂t + ( v · ∇ ) v = − M · ∇ ∆ λλ (17)and ∂λ ∂t + ∇ ( λ v ) = 0 . (18)Obviously, the same equations could be obtained immediately from Eqs. (2) and (3) uponimposing the Newtonian limit. The term on the r.h.s. of Eq. (17) is often called ’quantumpressure’. It relies on the spatial derivatives of the field λ . Hence, the quantum pressure is10egligible, provided that the field λ is distributed smoothly in space, i.e., when the followinginequality is obeyed (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ∆ λλ M ( v · ∇ ) v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ L · M · v (cid:28) . Not surprisingly, we have arrived at our condition (14). Once it is fulfilled, we result withthe pressureless Euler equation describing the non-relativistic evolution of collisionless dustparticles. This is known to be plagued by caustic singularities. At later times, when inho-mogeneities of the field λ grow, the quantum pressure cannot be ignored anymore, and onegets a chance to avoid instabilities.While we omit metric perturbations in the present paper, including the gravitationalpotential Φ is straightforward. One makes the replacement, − ∂ M ∂x → − ∂ M ∂x + M Φ . At the level of Madelung equations, this amounts to adding the gradient −∇ Φ to the r.h.s.of Eq. (17).
The canonical complex scalar is manifestly free of caustic singularities. Nevertheless, it isinteresting to see, how the real caustics of PPF is reflected in the complete picture andelucidate the mechanism of the singularity avoidance. This we do in the present Subsectionby considering a tractable example.Since the appearance of caustics is independent on the number of spatial dimensions, wecan consider the 1-dimensional example. We also neglect the cosmological drag for simplicity,i.e., we set a ( t ) = 1. We start with the following initial configuration of the complex scalarΨ = ˜ λ ( x, t ) e iMt + iδ ˜ ϕ ( x,t ) . Here we explicitly assumed the ’cosmological’ background value for the normalized phase (cid:104) ˜ ϕ (cid:105) = M t . We choose sufficiently smooth initial conditions for the normalized amplitude ˜ λ and the phase perturbation,˜ λ = A exp (cid:18) − x L (cid:19) δ ˜ ϕ ≡ ˜ ϕ − M t = Bx L (cid:48) . (19)Here A and B are some arbitrary constants; the length scales L and L (cid:48) characterize the sizeof initial inhomogeneities. The same choice of initial conditions was made in Ref. [49], wherethe ultra-light axion in the Bose-Einstein condensate state was discussed. That choice isconvenient, because it corresponds to the integrable Gaussian profile for the field Ψ.11 tT s Figure 2: Characteristics of PPF for the initial velocity profile (20). All characteristics crossat the same time t = T s forming the so called perfect caustics.The velocity following from the initial distribution of the phase is given by, v = − dM dx δ ˜ ϕ ( x ) = − BxM L (cid:48) . (20)Such a velocity profile growing linearly with the coordinate results into the so called perfectcaustics [51]. Employing for an instant the analogy with dust, all the particles fall into thecenter ( x = 0) at the same time leading to the multivalued velocity nearby x = 0. See Fig. 2.Before moving on, let us comment on the shortcoming of our profile (20) choice. In thecase of PPF, the Euler equation with the initial condition (20) can be easily integrated out.The result reads, v = − xT s − t , (21)where T s is the constant of integration defined from the initial condition (20), i.e., T s = M L (cid:48) B .
We see that at times t → T s , the velocity blows up at each point x . This is, how-ever, not a physical singularity, as it stems from admitting infinite velocities in the ini-tial distribution (20). For a realistic smooth distribution, the solution (21) gets modified v → − xT s − t + O (cid:16) x [ T s − t ] (cid:17) . Hence, in the limit of interest, t → T s , it can be trusted only in12he vicinity of the point x = 0. To see the real singularity, one should instead consider thevelocity divergence, i.e., ∂v ( x = 0) = − T s − t . This is a trustworthy expression. It shows explicitly that the description in terms of PPFbreaks down at the finite time t = T s .The solution for the complex scalar is given by,Ψ = (cid:90) dkα ( k ) e ikx + i k M t + iMt . (22)Writing it in this form, we explicitly assume the non-relativistic limit. It is straightforwardto find the amplitude α ( k ) from the initial conditions for the field Ψ (or ˜ λ and ˜ ϕ ). It reads α ( k ) = A √− πi (cid:114) TM · exp (cid:18) − ik T M (cid:19) , where we introduced the notation T ≡ T − iT = M L L (cid:48) B L + L (cid:48) (cid:16) BL − iL (cid:48) (cid:17) . (23)Note that in the limit M → ∞ , the time T → T s . On the other hand, for any large, butfinite M , T is a complex quantity. This observation is at the core of solving the causticsingularity in the picture of the complex scalar.Substituting this amplitude into Eq. (22) and integrating over the momentum k , we getfinally Ψ = A · e iMt (cid:113) − tT · exp (cid:18) − iM x t − T ) (cid:19) . (24)The velocity v = − ∂ x ϕ is related to the field Ψ by v = − Ψ ∂ x Ψ − Ψ ∂ x Ψ M · | Ψ | . From Eq. (24), we get v = − x ( T − t )( T − t ) + T , (25)(cf. Appendix E of Ref. [49]). This expression as well as its derivatives is manifestly finite atall the times, as it should be. Notably, the velocity divergence, ∂v , is negative at t < T andflips the sign at t > T . Using the analogy with particles, this corresponds to the situation,when particle trajectories tend to cross, but experience the repulsive force about the time t (cid:39) T . The repulsion is exactly due to the presence of the quantum pressure in Eq. (17),13hich is non-zero for any finite M . On the other hand, in the limit M → ∞ , when T → λ given by, λ = A M (cid:12)(cid:12)(cid:12)(cid:113) − tT (cid:12)(cid:12)(cid:12) · exp (cid:18) − M x T [( t − T ) + T ] (cid:19) . The latter always remains finite contrary to the case of PPF. At the time t = 0, thic correctlymatches the initial condition (19) for the field ˜ λ . This serves as a simple cross-check of ourcalculations.Despite multiple simplifications considered in the present Section, we assume that ourexample correctly reflects the real picture: the would-be collapse time is promoted to thecomplex quantity. Hence, the actual instability never occurs in the real time. According to the classification of Section 2, shift-symmetric k-essence scenarios correspondto the self-interacting complex scalar. The general analytic solution is not available in thatcase. Therefore, comparing k-essence and the complex field is a rather challenging task. Thisis still doable in the homogeneous case—the main focus of the present Section. Regarding theinhomogeneous evolution, we will be satisfied with translating the statements of Section 3into the context of k-essence.The equation of motion for the self-interacting complex field is given by¨Ψ + 3 H ˙Ψ + 2 ∂V ( | Ψ | ) ∂ | Ψ | Ψ = 0 . (26)For an instant let us neglect the cosmic drag. The equation admits the simple oscillatorysolution, Ψ = Ae ± iωt , (27)where the frequency ω is given by ω = (cid:115) ∂V ( | Ψ | ) ∂ | Ψ | . (28)Namely, we pick the specific configuration of the field Ψ described by the particular frequencydependence. See Fig. 1.Switching to the realistic case of the expanding Universe is straightforward. Though | Ψ | is not constant anymore, it is a slowly varying function of time. Therefore, we can solve14q. (26) in the WKB approximation,Ψ = Aa / √ ω · e ± i (cid:82) ω ( t ) dt . To find the time dependence of the frequency ω , let us take the absolute value squared ofthe left and right hand sides of the solution above,2 (cid:115) ∂V∂ | Ψ | | Ψ | = A a . (29)Using this and given the potential V , one can find | Ψ | . Plugging the result into Eq. (28), weobtain the frequency ω time dependence.As a concrete example, consider the potential V (Ψ) = M | Ψ | / . In that case, thefrequency is given by ω = 1 a (cid:18) A M (cid:19) / , (30)and the solution for the field Ψ can be written as follows,Ψ ∝ a · e ± i (cid:82) ωdt . Such a profile of the field Ψ corresponds to radiation. To show this, consider the energydensity ρ ( t ) = 12 | ˙Ψ | + V (Ψ) . For the quartic potential V , it redshifts as ρ ( t ) ∝ a , what proves the statement. This conclusion perfectly matches the result obtained in thek-essence scenario. Indeed, the quartic potential stands for the Lagrangian of the form L ( X ) = X . This Lagrangian effectively describes the radiation, as it should be.The inhomogeneous evolution of the canonical complex scalar is obviously caustic free.Still, it is unclear, if it reproduces k-essence models in the limit of large M . The issue iscomplicated due to the presence of the self-interacting potential. Therefore, we will formulateour conclusions by exploiting the analogy with PPF. With inhomogeneities included, the freescalar retains the PPF-like behaviour. We assume that the same works for k-essence. Hence,our conjecture: The self-interacting complex scalar reproduces k-essence, given that it has afixed frequency dependence, namely its homogeneous profile satisfies
Ψ = Aa / √ ω · e ± i (cid:82) ω ( t ) dt t → . λ and ˜ ϕ grow, the discrepancy between k-essence and the complex scalar becomes large. In particular, while the description in termsof k-essence breaks down at some point, no actual singularity occurs in the complete picture. Sub-/superluminality
Let us now comment on another pesky property of the k-essence:its perturbations exhibit the sub-/superluminality. The sound speed squared of k-essenceperturbations propagating in the preferred background is given by [4] c s = (cid:18) X L XX L X (cid:19) − . Taking again the Lagrangian
L ∝ X , we reproduce the standard result c s = , as it shouldbe in the case of radiation.For positive X and L X , the sign of L XX defines, if the sound speed is subluminal ( L XX >
0) or superluminal ( L XX < λ and the phase ˜ ϕ , makesthe property of luminality less transparent. The phase ˜ ϕ has a non-canonical kinetic term,which relies on the background value of the amplitude ˜ λ . Hence, for generic backgrounds, theemergence of sub-/superluminality is inevitable. We get back to the conventional luminalityupon switching to the canonical variables. In the present work, we pursued the unified completion of pressureless perfect fluid (PPF)and the shift-symmetric k-essence scenarios. In Section 2, we showed that they belong tothe same class of models involving two scalars. This class can be easily completed by meansof the unique complex field. We derived our main conclusions in Section 3 by exploiting thetractable example of the free massive scalar, which was our main reference point. Despite thesimplicity, this describes the physically interesting model—PPF. In Section 4, we generalizedthe discussion to the case of k-essence.One of our conclusions concerns the mechanism of caustic singularity avoidance. Weobserved that the PPF collapse time is promoted to the complex number in the completepicture. Hence, the real time evolution always remains smooth in the case of the canonicalscalar field, as it should be. This observation may have applications beyond the scope ofthe present research. Indeed, in the non-relativistic limit, the fixed frequency branch of thecomplex scalar reduces to the quantum mechanical wave function obeying the Schroedingerequation. The latter is often used to model collisionless particles without using N-bodysimulations. Finally, the ultra-light axion is described by the scalar field at sufficiently small16cales (still comparable with the size of halos). In all those cases, the mechanism of causticavoidance discussed in Subsection 3.4 is applied.Finally, we would like to point out several open issues. First, in the inflationary Uni-verse the amplitude of the scalar Ψ rapidly tends to zero with a high accuracy. In thissituation, the complex scalar represents just a collection of heavy particles above the trivialvacuum. Instead, we are interested in the non-trivial classical configuration of the complexfield shown in Fig. 1. However, this non-trivial configuration can be generated, if we ad-mit a slight breaking of the shift-symmetry (or, equivalently, U (1)-symmetry). The lattercan be achieved, e.g., by coupling the phase ˜ ϕ to the matter fields. More worrisome is ourassumption about tuned initial conditions for the complex scalar. Recall, that the lattershould have a particular frequency dependence. This may strongly constrain the mechanismof generating the field Ψ in the early Universe. We plan to get back to these issues in thefuture. Acknowledgments
We are indebted to Gilles Esposito-Farese for useful discussions.EB acknowledges support from the research program, Programme national de cosmologieet Galaxies of the CNRS/INSU, France, from the project DEFI InFIniTI 2017 and fromRussian Foundation for Basic Research Grant No. RFBR 15-02-05038.
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